 Continuous maps between cofinite and tail topologies

For a map $f:\mathbb{N}\to \mathbb{N}$ from the cofinite topology to the tail topology on the natural numbers, continuity is equivalent to every set $\{n:f(n)\ge k\}$ being empty or cofinite. If the image is unbounded, every such set is nonempty and cofinite, so $f(n)\to \infty$. If it is bounded, its maximum $M$ exists, and the inverse image of the tail at $M$ forces $f(n)=M$ eventually, with $f(n)\le M$ always. Conversely either condition makes every tail preimage empty or cofinite.